document.write( "Question 1052676: Solve the inequality.
\n" ); document.write( "x^4 - 11x^3 + 18x^2 > or equal to 0 \n" ); document.write( "
Algebra.Com's Answer #668064 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "x⁴ - 11x³ + 18x² ≥ 0\r\n" );
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document.write( "Factor out x²\r\n" );
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document.write( "x²(x² - 11x + 18) ≥ 0\r\n" );
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document.write( "Factor the trinomial in the parentheses:\r\n" );
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document.write( "x²(x - 2)(x - 9) ≥ 0\r\n" );
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document.write( "The critical numbers are 0, 2 and 9.\r\n" );
document.write( "They are also solution because when\r\n" );
document.write( "substituted for x the cause the left\r\n" );
document.write( "side to equal 0.\r\n" );
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document.write( "So we put those on a number line with\r\n" );
document.write( "a darkened circle:\r\n" );
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document.write( "----------☻-----☻--------------------☻---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12\r\n" );
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document.write( "We choose easiest test values in between those\r\n" );
document.write( "darkened circles.\r\n" );
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document.write( "for the interval x < 0  we choose -1  and substitute\r\n" );
document.write( "it in the inequality:\r\n" );
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document.write( "     x²(x - 2)(x - 9) ≥ 0\r\n" );
document.write( "(-1)²(-1 - 2)(-1 - 9) ≥ 0\r\n" );
document.write( "           1(-3)(-10) ≥ 0 \r\n" );
document.write( "                   30 ≥ 0\r\n" );
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document.write( "That's true so we shade the interval x < 0\r\n" );
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document.write( "-3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12\r\n" );
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document.write( "for the interval 0 < x < 2  we choose test value 1 \r\n" );
document.write( "and substitute it in the inequality:\r\n" );
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document.write( "     x²(x - 2)(x - 9) ≥ 0\r\n" );
document.write( "   (1)²(1 - 2)(1 - 9) ≥ 0\r\n" );
document.write( "            1(-1)(-8) ≥ 0 \r\n" );
document.write( "                   30 ≥ 0\r\n" );
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document.write( "That's true so we shade the interval x < 0\r\n" );
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document.write( "<=========☻=====☻--------------------☻---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12\r\n" );
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document.write( "We might as well delete the darkened circle at 0\r\n" );
document.write( "since now we know it is shaded anyway:\r\n" );
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document.write( "<===============☻--------------------☻---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12\r\n" );
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document.write( "for the interval 2 < x < 9  we choose test value 3 \r\n" );
document.write( "and substitute it in the inequality:\r\n" );
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document.write( "     x²(x - 2)(x - 9) ≥ 0\r\n" );
document.write( "   (3)²(3 - 2)(3 - 9) ≥ 0\r\n" );
document.write( "             9(1)(-6) ≥ 0 \r\n" );
document.write( "                  -54 ≥ 0\r\n" );
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document.write( "That's false so we do not shade the interval 2 < x < 9.\r\n" );
document.write( "So we still have:\r\n" );
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document.write( "-3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12\r\n" );
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document.write( "for the interval x > 9  we choose 10  and substitute\r\n" );
document.write( "it in the inequality:\r\n" );
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document.write( "     x²(x - 2)(x - 9) ≥ 0\r\n" );
document.write( "(10)²(10 - 2)(10 - 9) ≥ 0\r\n" );
document.write( "            100(8)(1) ≥ 0 \r\n" );
document.write( "                  100 ≥ 0\r\n" );
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document.write( "That's true so we shade the interval x > 9.\r\n" );
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document.write( "-3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12\r\n" );
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document.write( "That's the number line graph.  The interval\r\n" );
document.write( "notation for that is:\r\n" );
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document.write( "Edwin
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